A line perpendicular to the line segment joining the points A(1,0) and B(2,3), divides it at C in the ratio of 1:3. Then the equation of the line is

To determine the equation of the line that is perpendicular to the line segment joining the points \(A(1,0)\) and \(B(2,3)\) and divides it at \(C\) in the ratio of 1:3, we can follow these steps:

### Step 1: Determine the coordinates of point \(C\)
Given that \(C\) divides the segment \(AB\) in the ratio \(1:3\), we can use the section formula. The coordinates of \(C(x, y)\) are given by:
\[ x = \frac{mx_2 + nx_1}{m+n} \]
\[ y = \frac{my_2 + ny_1}{m+n} \]

Here, \(A(x_1, y_1) = (1, 0)\), \(B(x_2, y_2) = (2, 3)\), \(m = 1\), and \(n = 3\).

So,
\[ x = \frac{1 \cdot 2 + 3 \cdot 1}{1+3} = \frac{2 + 3}{4} = \frac{5}{4} \]
\[ y = \frac{1 \cdot 3 + 3 \cdot 0}{1+3} = \frac{3 + 0}{4} = \frac{3}{4} \]

Thus, the coordinates of point \(C\) are \(\left(\frac{5}{4}, \frac{3}{4}\right)\).

### Step 2: Determine the slope of \(AB\)
The slope of the line segment \(AB\) is given by:
\[ m_{AB} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 0}{2 - 1} = 3 \]

### Step 3: Determine the slope of the perpendicular line
The slope of a line perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of \(AB\) is \(3\), the slope of the perpendicular line is:
\[ m_{\perpendicular} = -\frac{1}{3} \]

### Step 4: Determine the equation of the perpendicular line
The perpendicular line passes through the point \(C\left(\frac{5}{4}, \frac{3}{4}\right)\) and has a slope of \(-\frac{1}{3}\).

Using the point-slope form of the equation of a line:
\[ y - y_1 = m(x - x_1) \]

Substitute \(y_1 = \frac{3}{4}\), \(x_1 = \frac{5}{4}\), and \(m = -\frac{1}{3}\):
\[ y - \frac{3}{4} = -\frac{1}{3}\left(x - \frac{5}{4}\right) \]

### Step 5: Simplify the equation
Multiply through by 12 to clear the fractions:
\[ 12 \left(y - \frac{3}{4}\right) = 12 \left(-\frac{1}{3} \left(x - \frac{5}{4}\right)\right) \]
\[ 12y - 9 = -4(x - \frac{5}{4}) \]
\[ 12y - 9 = -4x + 5 \]
\[ 12y = -4x + 14 \]
\[ 4x + 12y = 14 \]

Thus, the equation of the line is:
\[ 4x + 12y = 14 \]

Would you like further details or have any questions?

Here are some questions you might consider next:
1. How do you find the equation of a line given two points?
2. What is the section formula in coordinate geometry?
3. How do you find the slope of a line perpendicular to a given line?
4. How do you convert the point-slope form to the general form of a line equation?
5. What is the importance of the slope in determining the orientation of a line?

**Tip:** Always remember that the slope of a perpendicular line is the negative reciprocal of the original line's slope.

Explore the detailed solution to the problem of finding the equation of a line perpendicular to the line segment joining points A(1,0) and B(2,3), dividing it at C in the ratio of 1:3. Learn about coordinate geometry concepts, including the section formula and point-slope form of a line equation.

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